\newproblem{lay:6_6_5}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 6.6.5}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
	Let $X$ be the design matrix used to find the least-squares line to fit data $(x_1,y_1)$, $(x_2,y_2)$, ..., $(x_n,y_n)$. Use a theorem
	in Section 6.5 to show that the normal equations have a unique solution if and only if the data include at least two points with different
	$x$-coordinates.
}{
   % Solution
	Theorem 6.5.15 states that if the columns of $A$ are linearly independent, then $A$ can be factorized as $A=QR$ and the least-squares solution
	of the problem $A\mathbf{x}=\mathbf{b}$ is unique and given by $\hat{\mathbf{x}}=R^{-1}Q^T\mathbf{b}$.
	
	If the data points do not have two different $x$-coordinates, then the design matrix of the least-squares will be of the form
	\begin{center}
		$A=\begin{pmatrix} 1 & x_1 \\ 1 & x_1 \\ ... \\ 1 & x_1 \end{pmatrix}$
	\end{center}
	It can be easily seen that its two columns are not linearly independent because $\mathbf{a}_2=x_1\mathbf{a}_1$.
}
\useproblem{lay:6_6_5}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}

